Comprehensive Trigonometry Guide: Formulas, Laws, and Identities

This document provides a thorough introduction to trigonometry, beginning with the basics of triangles and angles, and then delving into the core trigonometric functions: sine, cosine, and tangent. It explains the unit circle, positive and negative values, and the conversion between degrees a...Read More

• Trigonometry ... is all about triangles.
• A triangle has three sides and three angles
• The three angles always add to 180°

Right Angled Triangles
• A right angled triangle is (you guessed it), a
triangle that
has a right angle (90°) in it.
• The little square in the corner tells us that it is
a right angled triangle
(I also put 90°, but you don't need to!)

Names
And we give names to each side:
 Adjacent is adjacent (next to) to the angle θ
 Opposite is opposite the angle θ
 the longest side is the Hypotenuse

"Sine, Cosine and Tangent"
• Trigonometry is good at find a missing side or
angle in a triangle.
• The special functions Sine, Cosine and
Tangent help us!
• They are simply one side of a triangle divided
by another.

For any angle "θ":
Sine Function:
sin(θ) = Opposite / Hypotenuse Cosine Function:
cos(θ) = Adjacent / Hypotenuse Tangent Function:
tan(θ) = Opposite / Adjacent (Sine, Cosine and Tangent are
often abbreviated to sin, cos and tan.)

Example
• What is the sine of 35°?
• Using this triangle (lengths are only to one
decimal place):
• sin(35°) = Opposite / Hypotenuse = 2.8/4.9 =
0.57...

Unit Circle
• It is a circle with a radius of 1 with its center at
0.
• Because the radius is 1, we can directly
measure sine, cosine and tangent.

Positive and Negative Values

Degrees and Radians
• Angles can be in Degrees or Radians. Here are
some examples:
Angle Degrees Radians
Right Angle 90° π/2
Straight Angle 180° π
Full Rotation 360° 2π

Repeating Pattern
• Because the angle is rotating around and
around the circle the Sine, Cosine and Tangent
functions repeat once every full rotation.
• When we need to calculate the function for an
angle larger than a full rotation of 2π (360°)
we subtract as many full rotations as needed
to bring it back below 2π (360°):

Odd/Even Identities
• 1. sin (-x) = -sin x
• 2. cos (-x) = cos x
• 3. tan (-x) = -tan x
• 4. cosec (-x) = -cosec x
• 5. sec (-x) = sec x
• 6. cot (-x) = -cot x

Cofunction Identities - radians
Periodicity Identities - radians
Periodicity Identities - degrees
Cofunction Identity - radians

Example
• what is the cosine of 370°?
• 370° is greater than 360° so let us subtract 360°
• 370° − 360° = 10°
• cos(370°) = cos(10°) = 0.985 (to 3 decimal places)
• And when the angle is less than zero, just add full rotations.
• what is the sine of −3 radians?
• −3 is less than 0 so let us add 2π radians
• −3 + 2π = −3 + 6.283 = 3.283 radians
• sin(−3) = sin(3.283) = −0.141 (to 3 decimal places)

Solving Triangles
• Find the Missing Angle "C“
• Angle C can be found using angles of a triangle
add to 180°
• So C = 180° − 76° − 34° = 70°

Other Functions (Cotangent, Secant,
Cosecant)
• Similar to Sine, Cosine and Tangent, there are three
other trigonometric functions which are made by
dividing one side by another:
• Cosecant Function:
• cosec(θ) = Hypotenuse / Opposite Secant Function:
• sec(θ) = Hypotenuse / Adjacent Cotangent Function:
• cot(θ) = Adjacent / Opposite

The Law of Sines
• The Law of Sines (or Sine Rule) is very useful
for solving triangles:
• It works for any triangle:
• a, b and c are sides.
• A, B and C are angles.
• (Side a faces angle A,
side b faces angle B and
side c faces angle C).

The Law of Cosines
• The Law of Cosines (also called the Cosine
Rule) is very useful for solving triangles:
• It works for any triangle:
• a, b and c are sides.
• C is the angle opposite side c

Exterior Angle Theorem
• For a triangle:
• The exterior angle d equals the angles a plus
b.
• The exterior angle d is greater than angle a, or
angle b.

Value Table

Formulas
• Pythagorean Identities
• Addition Formulas

Formulas
• Subtraction formulas
• Double Angle Formulas

Formulas
• Half-Angle Formulas
• Product Formulas

Formulas
• Factoring Formulas

Quotient Properties:
Reciprocal Properties:

Applications